An Introduction to Finsler Geometry by Xiaohuan Mo

By Xiaohuan Mo

This introductory booklet makes use of the relocating body as a device and develops Finsler geometry at the foundation of the Chern connection and the projective sphere package. It systematically introduces 3 sessions of geometrical invariants on Finsler manifolds and their intrinsic family members, analyzes neighborhood and international effects from vintage and glossy Finsler geometry, and offers non-trivial examples of Finsler manifolds pleasurable various curvature stipulations.

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61) It means that Chern connection is almost compatible with metric g. 62) An Introduction to Finsler Geometry 34 Proof. 63) and k ufvf - unkVin =5ik-^Fyi. 62). jUas Proof. 53), we obtain the following - F2FyiyJykgks. 60) K% = -vtaF2Fyiyjykgkl. 64) we have K^uas = ysytgtiFyiyjykgkl - F2Fyly]ykgks = -ysFyiyJ F2Fyiyjykgks. 4 We can rewrite the formula for K^uas KijUa Proof. 18) we have 9iJFj = | . 67). 53), we obtain the following + FFyiyjxk)ui + (ysFyiyj + F \yk X \--p5nl{Gksn Fyiyjyrgrs) 1 - Fysxk + Fykxs) — -Ul (Gskn ~ Fykx, + Fy,xk)].

6) Denote the horizontal left of y by y. 10) Covariant Differentiation 41 Proof. 10). 2 smooth function. (4-11) Le£ c(t) 6e a geodesic on (M,F), Then, on (x,y), we have Fdf = — [f{c{i))]t=ou mod ua, and f : SM un where c(0) = y. Proof. 7) y(f) = Jt[f(c{t))}t=o. e. the Hilbert form, by " • ". For example, we have / := f\n and Hija := Hija\n, ect. Note that / : SM —» R is a function and Hija is a three order tensor. 3 Landsberg curvature Let Hiia be the Cartan tensor of {M,F)(ci. 57)) . Put —Hap7 and L := ^^ Laf3

Similarly, set •Ja •=z Va\n = Vet where r\ = ^,ar]aea denote the Cartan form. We call Jawa Landsberg curvature and denote it by J. 3 if J = 0. 4([Shen, 1994]) A Finsler manifold is of weak Landsberg type if and only if all projective spheres in the projective sphere bundle are minimal. Let Vol(x) denote the volume of the unit Finsler sphere ([Bao and Chern, 1996]) {y£TxM\F(x,y) = l}. For functions a1 — ai{x), we have where n I dV := y/det{gij) ^ {-\)k-1ykdy1 A • • • A dyk A • • • A dyn. 5([Bao and Chern, 1996]) manifold.

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